The use of high speed rotating machinery is of such fundamental importance in modern industry that the operating speed and efficiency of many processes are defined by the capabilities of rotating equipment. In many cases maximum operating speed is limited by the onset of serious vibrations caused by imbalance of the rotor or rotating portion of the machine, and bearing wear can be a serious maintenance issue. If the machine is operating at a rotational speed well below the first flexural critical speed of the rotor shaft, the shaft can be considered to be stiff, in which case the imbalance can be described as a displacement and/or misalignment of the principal axis of inertia of the rotating mass relative to the axis of rotation. This imbalance is the result of unequal mass distribution about the axis of rotation. Balancing in the case of a rigid shaft can be accomplished by the so called "two-plane" or dynamic balancing procedure. At higher rotation speeds, shaft deformation becomes a factor, and the shaft must be considered as being flexible. The balancing procedure in the flexible-shaft situation is termed flexible-rotor balancing or "modal balancing". Balancing of either rigid and flexible rotors is accomplished by the placement of compensating balance masses along or about the rotor. The size and placement of these balance masses is usually determined by testing the rotor using a balancing machine, as described in chapter 39 of Shock & Vibration Handbook, Third Edition, 1988, edited by Cyril M. Harris, published by McGraw-Hill Book Company, New York. For the purpose of understanding the invention, the conventional process of testing and balancing the rotor of a machine when the machine is out of service and its rotor is placed on a balancing machine, as described in the abovementioned Handbook, is referred to as "manual" balancing, as contrasted with "automatic" balancing, which is a continuous adjustment process which takes place while the rotor is in service, implemented by some type of negative feed-back control system. Manual balancing is effective in dealing with rigid rotor vibrations, i.e. those vibrations which do not involve significant shaft flexion. This is because the rigid rotor vibration is the direct result of an initial asymmetry of mass distribution, and the asymmetry can be detected and corrected using a balancing machine. However, manual balancing is relatively ineffective in dealing with flexible rotor vibrations, because when the rotor operates at speeds which cause significant shaft flexure, a destabilizing positive feedback process becomes dominant in determining rotor dynamics. A very small initial imbalance becomes magnified in the flexible-rotor case because the shaft deforms in response to the net centrifugal force, and the deformation causes additional imbalance. This positive feedback process cannot be influenced by manual balancing, because manual balancing affects only the initial mass distribution and cannot respond to shaft flexure. On the other hand, automatic balancing, which is a control process, provides negative feedback which can compensate for the destabilizing effects of the positive feedback attributable to shaft flexure. With automatic balancing, the rotor can be operated at significantly higher speeds, and with less bearing wear, than without automatic balancing. However, prior art automatic balance systems are costly and complex. A prior art automatic balancing system might require electronic sensors to detect the imbalance, analog and/or digital electronic circuitry to process the imbalance input signals, and actuators to reposition the balance masses as required to implement the control effect. Improved automatic balancing systems are desired.